Physics Letters B 786 (2018) 100–105
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Correlations among elastic and inelastic cross-sections and slope
parameter
A.P. Samokhin
A.A. Logunov Institute for High Energy Physics of NRC “Kurchatov Institute”, Protvino, 142281, Russian Federation
a r t i c l e i n f o a b s t r a c t
Article history:
Received
24 August 2018
Accepted
15 September 2018
Available
online 19 September 2018
Editor:
B. Grinstein
Keywords:
High
energy pp interaction
Unitarity
condition
Total
cross-section
Elastic
cross-section
Inelastic
cross-section
Slope
parameter
We discuss the unitarity motivated relations among the elastic cross-section, slope parameter and
inelastic cross-section of the high energy pp interaction. In particular, the MacDowell-Martin unitarity
bound is written down in another form to make a relation between the elastic and inelastic quantities
more transparent. On the basis of an unitarity motivated relation we argue that the growth with energy
of the elastic to total cross-section ratio is a consequence of the increasing with energy of the inelastic
interaction intensity. The latter circumstance is an underlying reason for the acceleration of the slope
parameter growth, for the slowing of the growth of the elastic to total cross-section ratio and for other
interesting phenomena, which are observed in the TeV energy range. All of this confirms the old idea
that the elastic scattering is a shadow of the particle production processes.
© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
A growth with energy of the pp total cross-section σ
tot
(s) =
σ
el
(s) +σ
inel
(s) is due to that of the elastic σ
el
(s) and the inelastic
σ
inel
(s) cross-sections [1–7]. If the growth of σ
inel
(s) =
N(s)
n, inel
σ
n
(s)
can be formally attributed to a huge number of open inelastic
channels N(s) [8], the underlying reasons of the σ
el
(s) growth
are unknown. On the other hand, the unitarity condition relates
the properties of the elastic scattering amplitude with the contri-
bution
from the inelastic channels and the elastic scattering can
therefore be considered as a shadow of the particle production
processes [9]. In other words, due to unitarity there are some
correlations between behaviour of the characteristics of the elas-
tic
and inelastic scattering. Indeed, the MacDowell-Martin unitarity
bound [10]gives such a relation among the total cross-section, the
elastic cross-section and the slope parameter of the imaginary part
of the elastic scattering amplitude
B
I
(s) ≡2[
d
dt
ln
|Im T (s, t)|]
t=0
≥
σ
2
tot
(s)
18πσ
el
(s)
.
(1)
E-mail address: samokhin@ihep.ru.
In the present note we rewrite this inequality in another form to
make a relation between the elastic and inelastic quantities more
transparent.
According
to the optical theorem (which is a consequence of
the unitarity condition) the elastic differential cross-section at zero
value of the square of the four-momentum transfer, t, is related to
the total cross-section as
dσ
dt
|
t=0
=
σ
2
tot
(s)(1 + ρ
2
(s))
16π
, ρ(s) =
Re T (s, 0)
Im T (s, 0)
.
(2)
The slope of the forward diffraction peak, B(s ), and (dσ /dt)
t=0
are
determined experimentally by extrapolation of the nuclear elastic
scattering differential cross-section data at small values of t to the
forward direction t =0using the exponential form
dσ
dt
=
dσ
dt
|
t=0
exp(Bt). (3)
The experimental value of σ
tot
(s) is then calculated from Eq. (2)
(the
ratio of the real to the imaginary part of the elastic scattering
amplitude in the forward direction, ρ(s), is taken in this method
from the dispersion relations or from global model extrapolations).
If the local slope parameter, B(s, t) = d ln(dσ /dt)/dt, is approx-
imately
equal to B(s) in the essential for the value of integral
σ
el
(s) =
dt(dσ /dt) region 0 ≤|t| ≤|t
0
|, where |t
0
| ≈ 0.4GeV
2
,
the elastic cross-section is given by the following formula [11]
https://doi.org/10.1016/j.physletb.2018.09.032
0370-2693/
© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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